© Stanisław Jaworski. Article available under the CC BY-SA 4.0 licence
ARTICLE
ABSTRACT
The estimation of the fraction of a population with a stigmatizing characteristic is the issue that this study attempts to address. In this paper the nonrandomized response model proposed by Tian et al. (2007) is considered. The exact confidence interval (CI) for this fraction is constructed. The optimal sample size for obtaining the CI of a given length is also derived. In order to estimate the proportion of the population with a stigmatizing characteristic, we explore the nonrandomized response model proposed by Tian et al. (2007). The prevalent approach to constructing a CI involves applying the Central Limit Theorem. Unfortunately, such CIs fail to consistently maintain the prescribed confidence level, contradicting the Neyman (1934) definition o f C Is. I n t his p aper, w e p resent t he c onstruction o f a n e xact CIs for this proportion, ensuring adherence to the designated confidence l evel. T he l ength of the proposed CI depends on both the given probability of a positive response to a neutral question and the sample size. For these CIs, the probability of a positive response to a neutral question is established in relation to the provided limit on the privacy protection of the interviewee. Additionally, we derive the optimal sample size for obtaining a CI of a given length.
KEYWORDS
sensitive questions, nonrandomized response model, exact confidence interval.
REFERENCES
Clopper, C. J., Pearson, E. S., (1934). The Use of Confidence or Fiducial Limits Illustrated in the Case of the Binomial. Biometrika, 26(4), pp. 404–413.
Franklin, L. A., (1989). Randomized response sampling from dichotomous populations with continuous randomization. Survey Methodology, 15, pp. 225–235.
Greenberg, B. G., Abul-Ela, A.-L. A. and Horvitz, D. G., (1969). The Unrelated Question Randomized Response Model: Theoretical Framework. Journal of the American Statistical Association, 64, pp. 520–539.
Groenitz, H., (2014). A new privacy-protecting survey design for multichotomous sensitive variables. Metrika, 77, pp. 211–224.
Horvitz, D. G., Shah, B. V., Simmons and W. R., (1967). The Unrelated Question Randomized Response Model. in Proceedings of the Social Statistics Section. American Statistical Association, pp. 65–72.
Jaworski, S., Zieliński, W., (2023). The Optimal Sample Size in the Crosswise Model for Sensitive Questions. Applicationes Mathematicae, 50(1), pp. 2–34.
Kuk, A. Y. C., (1990). Asking Sensitive Question Indirectly. Biometrika, 77, pp. 436–438.
Liu, Y., Tian, G.-L., (2014). Sample size determination for the parallel model in a survey with sensitive questions. Journal of the Korean Statistical Society, 43(2), pp. 235–249. doi: 10.1016/j.jkss.2013.08.002.
Neyman, J., (1934). On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection. Journal of the Royal Statistical Society, 97, pp. 558–625.
Qiu, S.-F., Zou, G. Y. and Tang, M.-L., (2014). Sample size determination for estimating prevalence and a difference between two prevalences of sensitive attributes using the non-randomized triangular design. Computational Statistics & Data Analysis, 77, pp. 157–169. doi: 10.1016/j.csda.2014.02.019.
Raghavarao, D., (1978). On an Estimation Problem in Warner’s Randomized Response Technique. Biometrics. [Wiley, International Biometric Society], 34(1), pp. 87–90. Available at: http://www.jstor.org/stable/2529591 (Accessed: 1 July 2022).
Tan, M., Tian, G. L. and Tang, M. L., (2009). Sample Surveys With Sensitive Questions: A Non-Randomized Response Approach. The American Statistician, 63, pp. 9–16.
Tian, G. L., Yu, J. W., Tang, M. L. and Geng, Z., (2007). A New Nonrandomized Model for Analyzing Sensitive Questions with Binary Outcomes. Statistics in Medicine, 26, pp. 4238–4252.
Tian, G. L., Tang, M. L., Liu, Z., Tan, M. and Tang, N. S., (2011). Sample Size Determination for the Non-Randomised Triangular Model for Sensitive Questions in a Survey. Statistical Methods in Medical Research, 20, pp. 159–173.
Tian, G. L., (2014). A New Non-Randomized Response Model: The Parallel Model. Statistica Neerlandica, 68(4), pp. 293–323.
Warner, S. L., (1965). Randomized Response: A Survey Technique for Eliminating Evasive Answer Bias. Journal of the American Statistical Association, 60, pp. 63–69.
Yu, J.W., Tian, G. L. and Tang, M. L., (2008). Two New Models for Survey SamplingWith Sensitive Characteristic: Design and Analysis. Metrika, 67(3), pp. 251–263.